Posted in数学 物理学

物理学+数学：旋量简介

Posted in数学, 物理学

知乎的回答：

Clifford代数

和Clifford代数联系紧密的是四元数 [公式] ，它是 [公式] 上的4维线性空间，在四元数的乘法下成为代数。取一组基 [公式] ，那么四元数的乘法按 [公式] 定义。四元数和 [公式] 群有紧密联系。通常，由Pauli矩阵 [公式] ， [公式] 和 [公式] ，可以取 [公式] 为Lie代数 [公式] 的一组基。从而 [公式] 中的元素都可以写成 [公式] ，其中 [公式] ， [公式] ， [公式] 且 [公式] 。易验证 [公式] 是单位阵。应用指数映射， [公式] 的单参数子群为 [公式] ，由于矩阵Lie群可以展开为Taylor级数，所以有 [公式] ，从而 [公式] 的矩阵均形如 [公式] ，其中 [公式] ， [公式] 。易看出 [公式] ，所以实际上这四个参数构成了球面 [公式] ，即 [公式] 。四元数有忠实表示 [公式] 。令 [公式] ，[公式] ，那么四元数 [公式] 表示为 [公式] 。

令 [公式] 是代数，不妨将其上的二元运算看成乘法。称子代数 [公式] 是左理想，若 [公式] 和 [公式] 使得 [公式] 。若是右理想，则 [公式] 。若同时是左理想和右理想，则称它是双边理想。显然，交换环上定义的交换代数本身就是交换环，所以在给定双边理想后，可以按环论中的做法，引入商环。给定交换环上的环 [公式] 和其双边理想 [公式] ，定义等价关系 [公式] 若 [公式] 。这样可以直接记等价类为 [公式] 。所有等价类的集合再次构成环，其加法按 [公式] 定义，乘法按 [公式] 定义。称这个环是商环，记作 [公式] 。若 [公式] 是代数，那么是交换环，并且商环 [公式] 也是交换环，进而是代数。称代数 [公式] 是因子代数。我们要指出，外代数实际上就是张量代数的因子代数。对于有限维线性空间 [公式] ，令 [公式] 是只有下标的张量代数， [公式] 是所有形如 [公式] 的和构成的双边理想，其中 [公式] ， [公式] 。容易验证 [公式] 是双边理想，并且 [公式] 。为了说明这一点，注意到商环的定义，它实际上是让理想消灭了，那么我们理应让对称形式消灭，只留下交错形式。据定义，取 [公式] ，那么 [公式] ，从而 [公式] 。对于 [公式] ，有 [公式] ，若其中两个分量相同，则 [公式] ，这就表明确实只剩下了交错形式。注意到，虽然代数 [公式] 和理想 [公式] 都是无穷维的，但因子代数却是有限维的。

 

Posted in物理学 理论物理

物理学：黑洞信息悖论进展——The Black Hole Information Paradox Comes to an End【转载】

Posted in物理学, 理论物理

原文来自：The Black Hole Information Paradox Comes to an End | Quanta Magazine

The Most Famous Paradox in Physics Nears Its End

In a landmark series of calculations, physicists have proved that black holes can shed information, which seems impossible by definition. The work appears to resolve a paradox that Stephen Hawking first described five decades ago.

In a series of breakthrough papers, theoretical physicists have come tantalizingly close to resolving the black hole information paradox that has entranced and bedeviled them for nearly 50 years. Information, they now say with confidence, does escape a black hole. If you jump into one, you will not be gone for good. Particle by particle, the information needed to reconstitute your body will reemerge. Most physicists have long assumed it would; that was the upshot of string theory, their leading candidate for a unified theory of nature. But the new calculations, though inspired by string theory, stand on their own, with nary a string in sight. Information gets out through the workings of gravity itself — just ordinary gravity with a single layer of quantum effects.

This is a peculiar role reversal for gravity. According to Einstein’s general theory of relativity, the gravity of a black hole is so intense that nothing can escape it. The more sophisticated understanding of black holes developed by Stephen Hawking and his colleagues in the 1970s did not question this principle. Hawking and others sought to describe matter in and around black holes using quantum theory, but they continued to describe gravity using Einstein’s classical theory — a hybrid approach that physicists call “semiclassical.” Although the approach predicted new effects at the perimeter of the hole, the interior remained strictly sealed off. Physicists figured that Hawking had nailed the semiclassical calculation. Any further progress would have to treat gravity, too, as quantum.

That is what the authors of the new studies dispute. They have found additional semiclassical effects — new gravitational configurations that Einstein’s theory permits, but that Hawking did not include. Muted at first, these effects come to dominate when the black hole gets to be extremely old. The hole transforms from a hermit kingdom to a vigorously open system. Not only does information spill out, anything new that falls in is regurgitated almost immediately. The revised semiclassical theory has yet to explain how exactly the information gets out, but such has been the pace of discovery in the past two years that theorists already have hints of the escape mechanism.

“That is the most exciting thing that has happened in this subject, I think, since Hawking,” said one of the co-authors, Donald Marolf of the University of California, Santa Barbara.

“It’s a landmark calculation,” said Eva Silverstein of Stanford University, a leading theoretical physicist who was not directly involved.

You might expect the authors to celebrate, but they say they also feel let down. Had the calculation involved deep features of quantum gravity rather than a light dusting, it might have been even harder to pull off, but once that was accomplished, it would have illuminated those depths. So they worry they may have solved this one problem without achieving the broader closure they sought. “The hope was, if we could answer this question — if we could see the information coming out — in order to do that we would have had to learn about the microscopic theory,” said Geoff Penington of the University of California, Berkeley, alluding to a fully quantum theory of gravity.

What it all means is being intensely debated in Zoom calls and webinars. The work is highly mathematical and has a Rube Goldberg quality to it, stringing together one calculational trick after another in a way that is hard to interpret. Wormholes, the holographic principle, emergent space-time, quantum entanglement, quantum computers: Nearly every concept in fundamental physics these days makes an appearance, making the subject both captivating and confounding.

And not everyone is convinced. Some still think that Hawking got it right and that string theory or other novel physics has to come into play if information is to escape. “I’m very resistant to people who come in and say, ‘I’ve got a solution in just quantum mechanics and gravity,’” said Nick Warner of the University of Southern California. “Because it’s taken us around in circles before.”

But almost everyone appears to agree on one thing. In some way or other, space-time itself seems to fall apart at a black hole, implying that space-time is not the root level of reality, but an emergent structure from something deeper. Although Einstein conceived of gravity as the geometry of space-time, his theory also entails the dissolution of space-time, which is ultimately why information can escape its gravitational prison.

The Curve Becomes the Key

In 1992, Don Page and his family spent their Christmas vacation house-sitting in Pasadena, enjoying the swimming pool and watching the Rose Parade. Page, a physicist at the University of Alberta in Canada, also used the break to think about how paradoxical black holes really are. His first studies of black holes, when he was a graduate student in the ’70s, were key to his adviser Stephen Hawking’s realization that black holes emit radiation — the result of random quantum processes at the edge of the hole. Put simply, a black hole rots from the outside in.

 

Posted in物理学 理论物理 科普 量子引力

物理学：Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.

Posted in物理学, 理论物理, 科普, 量子引力
本文转载自：

Does Time Really Flow? New Clues Come From a Century-Old Approach to Math.

By Natalie Wolchover

April 7, 2020

The laws of physics imply that the passage of time is an illusion. To avoid this conclusion, we might have to rethink the reality of infinitely precise numbers.

If numbers cannot have infinite strings of digits, then the future can never be perfectly preordained.

Dave Whyte for Quanta Magazine

Strangely, although we feel as if we sweep through time on the knife-edge between the fixed past and the open future, that edge — the present — appears nowhere in the existing laws of physics.

Posted in物理学 理论物理

物理学：一文读懂量子纠缠【转载】

Posted in物理学, 理论物理

本文转载自知乎：一文读懂量（xiang）子（ai）纠（xiang）缠（sha）

英文原文1：Entanglement Made Simple

英文原文2：Your Simple (Yes, Simple) Guide to Quantum Entanglement

量子纠缠及其“多世界”诠释都带有一种神秘而迷人的光环。然而，这些都是，或者都应该是科学观点，它们都有实实在在的具体含义。在下面这篇文章中，我们将尽可能简单明了地为大家解释一下量子纠缠和多世界的概念。

纠缠：从经典迈入量子

量子纠缠经常被看作量子力学才独有的现象，但事实并不是这样。实际上，我们可以首先通过思考一个简单的非量子（或者“经典”）现象来考察纠缠，这是一种比较反传统的做法。这样可以让我们绕开量子论中纠缠的怪异之处来体会量子纠缠的精妙。

一个系统由两个子系统组成，纠缠发生在我们对系统的状态有部分了解的情况下。我们将子系统称之为c-on。“c”的意思是“经典的”，为了便于理解，我们把c-on看作蛋糕。

这里我们的蛋糕有两种形状，正方形或者圆形。那么两个蛋糕的总状态就有4种，它们分别是（方，方）（方，圆）（圆，方）（圆，圆）。下面两个表格给出了在四个状态中找到某一个状态的概率。

当我们不能通过一个蛋糕的信息来判断另一个蛋糕的状态时，我们称这两个子系统是独立的。我们的第一个表格就具有这种特性。即使我们知道第一个蛋糕是方的，我们仍然不知道另一个的形状。类似的，第二个子系统的形状并不能告诉我们关于第一个子系统形状的任何有用信息。

另一方面，如果一个蛋糕的信息可以增加我们对另一个蛋糕的认识，我们就说这两个蛋糕是纠缠的。第二个表格中的情况就表现出高度的纠缠。在这种情况中，如果我们已经知道第一个蛋糕是圆的，那么我们就知道第二个蛋糕一定也是圆形的。如果第一个蛋糕是方形的，第二个也是。当我们知道了第一个蛋糕的形状我们就能确定另一个蛋糕的形状。

Posted in物理学 理论物理

物理学：论动体的电动力学-英文版【转载】

Posted in物理学, 理论物理

转载自：http://www.fourmilab.ch/etexts/einstein/specrel/www/

pdf版本下载：PDF

还有Latex源代码，PostScript格式的下载： LaTeX Source, PostScript

ON THE ELECTRODYNAMICS
OF MOVING BODIES

By A. Einstein
June 30, 1905

It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.1 We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell’s theory for stationary bodies. The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

The theory to be developed is based—like all electrodynamics—on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

 

I. KINEMATICAL PART

§ 1. Definition of Simultaneity

Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good.2 In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the “stationary system.”

If a material point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates.

If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by “time.” We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, “That train arrives here at 7 o’clock,” I mean something like this: “The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events.”3

It might appear possible to overcome all the difficulties attending the definition of “time” by substituting “the position of the small hand of my watch” for “time.” And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch.

We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.

If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time” and a “B time.” We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A. Let a ray of light start at the “A time” $t_{\rm A}$from A towards B, let it at the “B time” $t_{\rm B}$ be reflected at B in the direction of A, and arrive again at A at the “A time” $t'_{\rm A}$.

In accordance with definition the two clocks synchronize if

\begin{displaymath}t_{\rm B}-t_{\rm A}=t'_{\rm A}-t_{\rm B}. \end{displaymath}

We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—

  1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
  2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

In agreement with experience we further assume the quantity

\begin{displaymath}\frac{2{\rm AB}}{t'_A-t_A}=c, \end{displaymath}

to be a universal constant—the velocity of light in empty space.

It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the stationary system.”